I can still measure some voltage on the output of either side leaving one of the leads unconnected.
Voltage is a 2-node quantity. \$V_{ab}\$ means the voltage from node \$a\$ (the red lead) to node \$b\$ (the black lead). Sometimes voltages are written with a single subscript, as in \$V_{a}\$. In this case it is assumed that the voltage is relative to a reference node. It can then be said that \$V_{ab}=V_{a}-V_{b}\$. When you leave one lead (say it is node \$b\$) disconnected, the node is in the air where the probe is. When you touch the lead in the are the voltage at that point changes. Voltages measured this way are meaningless.
The oscilloscope is a voltmeter with the black lead connected to ground. The probe is similar to the red lead on a DMM.
You must understand that the meter display indicates the voltage difference from the red lead to the black lead.
Is this due to some stray capacitance and can this be called 'phantom voltage'?
There is capacitance between the disconnected probe and every other conductive object nearby. There is also interwinding capacitance. The meter is high impedance so observing a voltage is not unreasonable.
There is no such phantom voltage. All voltages are real, even the ones that are meaningless.
When I increase the output voltage at first this 'phantom voltage' goes up (to ~20-30VAC), later to go down again (to ~5-10V). How can this be explained?
The capacitance of the disconnected lead is changing as you move around it or move it around.
When I connect a microwave transformer to the output of the variac. It's output pole still shows a voltage when I increase the input on the variac (still with one of the measuring leads unconnected),...
The voltage output is from one terminal (pole is not the correct term) to a second terminal. You cannot measure with one lead disconnected.
...but the mass of this transformer does not show any voltage.
Mass has nothing to do with with voltage.
Is there any significant difference between what is called primary and secondary on an isolation transformer or can they safely be reversed?
Sometimes yes, sometimes no. You will have to read the datasheet.
You seem to have a misunderstanding of the nature of voltage and how to measure it.
Imagine that I want to measure the length of a box. I have a string that has millimeter markings on it. I holder the string to the left edge of the box and read 135mm. I go away and come back and hold the string against the left edge and get 199mm. I repeat this over and over and get variety of measurements that don't make sense.
I call a friend over and explain the problem. I suspect that there is something wrong with the string. She notices that whenever I hold the string to the box , the zero und is left unconnected, then tells me to connect the zero end to the right side and hold the string to the left side at the same time. I now measure 160 mm consistently over several measurements. I also notice that if I place an arbitrary point on the right side, then subtract the value from the left side, I also get 160 mm.
Distance is a 2-point quantity. So is voltage.